AMS / MAA TEXTBOOKS VOL 23 Learning Modern Algebra From Early Attempts to Prove Fermat’s Last Theorem Al Cuoco and Joseph J. Rotman i i “book2” — 2013/5/24 — 8:18 — page i — #1 i i 10.1090/text/023 Learning Modern Algebra From Early Attempts to Prove Fermat’s Last Theorem i i i i i i “book2” — 2013/5/24 — 8:18 — page ii — #2 i i c 2013 by The Mathematical Association of America (Incorporated) Library of Congress Control Number: 2013940990 Print ISBN: 978-1-93951-201-7 Electronic ISBN: 978-1-61444-612-5 Printed in the United States of America Current Printing (last digit): 10987654321 i i i i i i “book2” — 2013/5/24 — 8:18 — page iii — #3 i i Learning Modern Algebra From Early Attempts to Prove Fermat’s Last Theorem Al Cuoco EDC, Waltham MA and Joseph J. Rotman University of Illinois at Urbana–Champaign Published and distributed by The Mathematical Association of America i i i i i i “book2” — 2013/5/24 — 8:18 — page iv — #4 i i Committee on Books Frank Farris, Chair MAA Textbooks Editorial Board Zaven A. Karian, Editor Matthias Beck Richard E. Bedient Thomas A. Garrity Charles R. Hampton John Lorch Susan F. Pustejovsky Elsa J. Schaefer Stanley E. Seltzer Kay B. Somers MAA TEXTBOOKS Bridge to Abstract Mathematics, Ralph W. Oberste-Vorth, Aristides Mouzakitis, and Bonita A. Lawrence Calculus Deconstructed:A Second Coursein First-Year Calculus, Zbigniew H. Nitecki Combinatorics: A Guided Tour, David R. Mazur Combinatorics: A Problem Oriented Approach, Daniel A. Marcus Complex Numbers and Geometry, Liang-shin Hahn A Course in Mathematical Modeling, Douglas Mooney and Randall Swift Cryptological Mathematics, Robert Edward Lewand Differential Geometry and its Applications, John Oprea Elementary Cryptanalysis, Abraham Sinkov Elementary Mathematical Models, Dan Kalman An Episodic History of Mathematics: Mathematical Culture Through Problem Solving, Steven G. Krantz Essentials of Mathematics, Margie Hale Field Theory and its Classical Problems, Charles Hadlock Fourier Series, Rajendra Bhatia Game Theory and Strategy, Philip D. Straffin Geometry Revisited, H. S. M. Coxeter and S. L. Greitzer Graph Theory: A Problem Oriented Approach, Daniel Marcus Knot Theory, Charles Livingston Learning Modern Algebra: From Early Attempts to Prove Fermat’s Last Theorem, Al Cuoco and and Joseph J. Rotman Lie Groups: A Problem-Oriented Introduction via Matrix Groups, Harriet Pollatsek Mathematical Connections: A Companion for Teachers and Others, Al Cuoco Mathematical Interest Theory, Second Edition, Leslie Jane Federer Vaaler and James W. Daniel i i i i i i “book2” — 2013/5/24 — 8:18 — page v — #5 i i Mathematical Modeling in the Environment, Charles Hadlock Mathematics for BusinessDecisionsPart 1: Probability and Simulation (electronic text- book), Richard B. Thompson and Christopher G. Lamoureux Mathematics for Business DecisionsPart 2: Calculus and Optimization (electronic text- book), Richard B. Thompson and Christopher G. Lamoureux Mathematics for Secondary School Teachers, Elizabeth G. Bremigan, Ralph J. Bremi- gan, and John D. Lorch The Mathematics of Choice, Ivan Niven The Mathematics of Games and Gambling, Edward Packel Math Through the Ages, William Berlinghoff and Fernando Gouvea Noncommutative Rings, I. N. Herstein Non-Euclidean Geometry, H. S. M. Coxeter Number Theory Through Inquiry, David C. Marshall, Edward Odell, and Michael Star- bird A Primer of Real Functions, Ralph P. Boas A Radical Approach to Lebesgue’s Theory of Integration, David M. Bressoud A Radical Approach to Real Analysis, 2nd edition, David M. Bressoud Real Infinite Series, Daniel D. Bonar and Michael Khoury, Jr. Topology Now!, Robert Messer and Philip Straffin Understanding our Quantitative World, Janet Andersen and Todd Swanson MAA Service Center P.O. Box 91112 Washington, DC 20090-1112 1-800-331-1MAA FAX: 1-301-206-9789 i i i i i i “book2” — 2013/5/24 — 8:18 — page vi — #6 i i i i i i i i “book2” — 2013/5/24 — 8:18 — page vii — #7 i i vii Per Micky: Tutto quello che faccio, lo faccio per te. i i i i i i “book2” — 2013/5/24 — 8:18 — page viii — #8 i i i i i i i i “book2” — 2013/5/24 — 8:18 — page ix — #9 i i Contents Preface xiii SomeFeaturesofThisBook . xiv ANotetoStudents .......................... xv ANotetoInstructors ... ........ ....... ....... xv Notation xvii 1 Early Number Theory 1 1.1 AncientMathematics ........ ....... ....... 1 1.2 Diophantus ........................... 7 GeometryandPythagoreanTriples . 8 TheMethodofDiophantus . 11 Fermat’sLastTheorem . 14 Connections:CongruentNumbers. 16 1.3 Euclid.............................. 20 GreekNumberTheory. 21 DivisionandRemainders . 22 LinearCombinationsandEuclid’sLemma . 24 EuclideanAlgorithm. 30 1.4 NineFundamentalProperties . 36 1.5 Connections........................... 41 Trigonometry ........................ 41 Integration .......................... 42 2 Induction 45 2.1 InductionandApplications . 45 UniqueFactorization. 53 StrongInduction. ........ ....... ....... 57 DifferentialEquations . 60 2.2 BinomialTheorem . ........ ....... ....... 63 Combinatorics ........................ 69 2.3 Connections........................... 73 AnApproachtoInduction . 73 FibonacciSequence . 75 3 Renaissance 81 3.1 ClassicalFormulas . 82 3.2 ComplexNumbers ....................... 91 ix i i i i i i “book2” — 2013/5/29 — 16:18 — page x — #10 i i x Contents AlgebraicOperations . 92 AbsoluteValueandDirection . 99 TheGeometryBehindMultiplication . 101 3.3 RootsandPowers ........ ....... ........ 106 3.4 Connections:DesigningGoodProblems . 116 Norms ............................ 116 PippinsandCheese . 118 Gaussian Integers: Pythagorean Triples Revisited . 119 EisensteinTriplesandDiophantus. 122 NiceBoxes.......................... 123 NiceFunctionsforCalculusProblems . 124 LatticePointTriangles. 126 4 Modular Arithmetic 131 4.1 Congruence ........................... 131 4.2 PublicKeyCodes ........ ....... ........ 149 4.3 CommutativeRings. 154 UnitsandFields ....... ....... ........ 160 SubringsandSubfields. 166 4.4 Connections:JuliusandGregory . 169 4.5 Connections:PatternsinDecimalExpansions . 177 RealNumbers ........................ 177 DecimalExpansionsofRationals . 179 PeriodsandBlocks. 182 5 Abstract Algebra 191 5.1 DomainsandFractionFields . 192 5.2 Polynomials........................... 196 PolynomialFunctions . 204 5.3 Homomorphisms ........ ....... ........ 206 ExtensionsofHomomorphisms . 213 Kernel,Image,andIdeals . 216 5.4 Connections:BooleanThings. 221 Inclusion-Exclusion . 227 6 Arithmetic of Polynomials 233 6.1 Parallels to Z .......................... 233 Divisibility.......................... 233 Roots............................. 239 GreatestCommonDivisors . 243 UniqueFactorization. 248 PrincipalIdealDomains . 255 6.2 Irreducibility .. ........ ....... ........ 259 RootsofUnity ........................ 264 6.3 Connections:LagrangeInterpolation . 270 7 Quotients, Fields, and Classical Problems 277 7.1 QuotientRings ......................... 277 7.2 FieldTheory........................... 287 Characteristics . 287 ExtensionFields . 289 i i i i i i “book2” — 2013/5/24 — 8:18 — page xi — #11 i i Contents xi AlgebraicExtensions . 293 SplittingFields. 300 ClassificationofFiniteFields . 305 7.3 Connections:Ruler–CompassConstructions . 308 Constructing Regular n-gons ................ 320 Gauss’sconstructionofthe17-gon . 322 8 Cyclotomic Integers 329 8.1 ArithmeticinGaussianandEisensteinIntegers . 330 EuclideanDomains . 333 8.2 PrimesUpstairsandPrimesDownstairs . 337 LawsofDecomposition . 339 8.3 Fermat’sLastTheoremforExponent3 . 349 Preliminaries... ........ ....... ....... 350 TheFirstCase ........................ 351 Gauss’sProofoftheSecondCase . 354 8.4 ApproachestotheGeneralCase . 359 Cyclotomicintegers . 360 Kummer,IdealNumbers,andDedekind. 365 8.5 Connections:CountingSumsofSquares . 371 AProofofFermat’sTheoremonDivisors . 373 9 Epilog 379 9.1 AbelandGalois... ........ ....... ....... 379 9.2 SolvabilitybyRadicals . 381 9.3 Symmetry............................ 384 9.4 Groups.............................. 389 9.5 WilesandFermat’sLastTheorem . 396 EllipticIntegralsandEllipticFunctions . 397 CongruentNumbersRevisited. 400 EllipticCurves. 404 A Appendices 409 A.1 Functions ............................ 409 A.2 EquivalenceRelations. 420 A.3 VectorSpaces .... ........ ....... ....... 424 BasesandDimension . 427 LinearTransformations . 435 A.4 Inequalities ........................... 441 A.5 GeneralizedAssociativity. 442 A.6 ACyclotomicIntegerCalculator . 444 EisensteinIntegers . 445 SymmetricPolynomials . 446 AlgebrawithPeriods. 446 References 449 Index 451 About the Authors 459 i i i i i i “book2” — 2013/5/24 — 8:18 — page xii — #12 i i i i i i i i “book2” — 2013/5/24 — 8:18 — page xiii — #13 i i Preface This book is designed for college students who want to teach mathematics in high school, but it can serve as a text for standard abstract algebra courses as well. First courses in abstract algebra usually cover number theory, groups, and commutative rings. We have found that the first encounter with groups is not only inadequate for future teachers of high school mathematics, it is also unsatisfying for other mathematics students. Hence, we focus here on number theory, polynomials, and commutative rings. We introduce groups in our last chapter, for the earlier discussion of commutative rings allows us to explain how groups are used to prove Abel’s Theorem: there is no generalization of the quadratic, cubic, and quartic formulas giving the roots of the general quintic polynomial. A modest proposal: undergraduate abstract algebra